On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free Poisson

TL;DR

This paper develops a method to compute the absolutely continuous part of the Brown measure for sums of freely independent selfadjoint operators, specifically when one has a free Poisson distribution, using matrix-valued subordination functions and a change of variables.

Contribution

It introduces a new parametrization approach for the Brown measure of $x + i y$, leveraging the matrix-valued subordination function and its inverse to derive an explicit density formula.

Findings

Derived an explicit formula for the Brown measure density.

Established conditions under which the formula applies.

Provided a new computational approach for free probability measures.

Abstract

Let $x,y$ be freely independent selfadjoint elements in a $W^{*}$-probability space, where $y$ has free Poisson distribution of parameter $p$. We pursue a methodology for computing the absolutely continuous part of the Brown measure of $x + i y$, which relies on the matrix-valued subordination function $\Omega$ of the Hermitization of $x + i y$, and on the fact that $\Omega$ has an explicitly described left inverse $H$. Our main point is that the Brown measure of $x + i y$ becomes more approachable when it is reparametrized via a certain change of variable $h : \mathcal{D} \to \mathcal{M}$, with $\mathcal{D}, \mathcal{M}$ open subsets of $\mathbb{C}$, where $\mathcal{D}$ and $h$ are defined in terms of the aforementioned left inverse $H$, and $\mathrm{cl} \,(\mathcal{M})$ contains the support of the Brown measure. More precisely, we find (with some conditions on the distribution of $x$, which have to be imposed for certain values of the parameter $p$) the following formula: \[ f(s + i \, t) =\frac{1}{4\pi}\left[\frac{2}{t}\left(\frac{\partial \alpha}{\partial s} +\frac{\partial \beta}{\partial t}\right)-\frac{2}{t}-\frac{2\beta}{t^2}\right], \ \ s + i \, t \in \mathcal{M}, \] where $f$ is the density of the absolutely continuous part of the Brown measure and the functions $\alpha, \beta : \mathcal{M} \to \mathbb{R}$ are the real and respectively the imaginary part of $h^{-1}$.